Can I just intercede here? I think BT is being misunderstood a bit somewhere on here. Here's a real-world analogy for BT.

You buy some electrical appliance, unbox it, untie the cable, and then you cannot seem to get the whole lot inside the box again no matter what you try.

Obviously this isn't BT. It's saying there's some operation (unboxing) which seems to magically increase the size of something. It hasn't, but you can fit the pieces back together again, and it feels tight, but the volume has gone up. Closer to BT would be if you have a mechanism with two plates with interlocking teeth, then move one half a click. They now occupy more space although nothing has changed. If the teeth were infinitely fine, you could do this and not see any gap.

I don't know if this helps, maybe not. But it's how I explain BT to undergraduates.

Below is my interpretation of the type of sets B-T Paradox is discussing. Simplified to the shape of the circle. I'm taking no position, what I am asking, is if my resulting statements are representative of what B-T Paradox is saying? I'm trying to discover where I ran off the rails in my reasoning. Care to respond?

I'll define a shape and call it a circle.

I will further say, that I can define that shape with 1 point and a origin, if the that shape is closed.

Furthermore, I consider the shape closed, if and only if, when I rotate the point about the origin, I end where I started.

I consider the area to be a finite and countable set of points, contained in that shape, when that shape is closed. Those finite points, from an origin, make up the points on a closed circle.

When the set is defined in this fashion, I can then construct a second circle, which shares a common point, such that when the point sweeps through the defined, interval with the stated condition for closure, that it sweeps both circles before becoming closed. And that both circles meet the conditions of closure. And both those circles are members of the original set.

I also state that when this condition is true, when I close the shape there is at least one set, which is not closed, that is the shape minus its origin. Which is an ?uncountable? set. Which is is also a member of the sets contained in the shape.

This follows from a condition of the definition. Both circles are closed only when the number of sweeps is n greater than or equal to 3.

End

The last merely says the second circle always closes first. I assume by symmetry this is also true for a sphere defined in the same manner.

doogly wrote:I could try to figure out all the ways in which you go off the rails, but the problem really seems to be that you are trying to lay down your own track.

You should not come up with your own definitions of things like "closed." Halmos' set theory book and Armstrong's topology are really nice and good places to get going, I would recommend these.

I might well at some point. But until then I can define in any way that makes it clear to me. Because at this point it is my best understanding. What I asked him to do is to test my model and tell me, if he wished to share, if the conceptual picture I gave, bounded in the fashion I bounded it, is equivilant to the B-T Paradox, and if not, how? He evidently teaches something where this comes up.

You aggravate me because you aren't helpful. It would be better if you didn't answer, and there is no requirement that you do so.

You just keep telling me you know better.

You're equivilant to a teacher and goes into a classroom and tells his students to, "Read the book!", and then leaves.

The student never truly learns it until she measures his/her knowledge against a known good source.

Their knowledge is defective, and they can't correct that knowledge, if they aren't first told, how it is wrong.

I have a map that says, There be Dragons here. Since the map is blank, whatever I think about the map is only exposed when I go somewhere, and somebody tells me that the place I wanted to go, isn't the place that I ended up.

I really don't have any reservations about your math skills as compared to mine, they're better. What I doubt is your skill as a teacher. If you don't want to teach idiots, don't, but please don't tell the idiots to read the book, it isn't very helpful. I'm an idiot, remember?

To be clearer, I'm calling your response an ad hominem. You're talking about me and not the question I asked.

If you haven't seen Vsauce's video about the B-T Paradox, I think you should. Even if you don't follow exactly how all of the "pieces" of the sphere are manipulated to produce two, it'll at least give you an understanding of how the pieces are constructed and why it doesn't at all resemble what your interpretation of the paradox seems to be.

A hotel with a countably infinite number of rooms, numbered with the positive integers in order

who has a son that wants to go into the same business. Making the Hilbert Hotel cost an infinite amount of money, and that in turn led to an infinite amount of taxes, so instead of making another one from scratch, he decides to give his son an infinite piece of his own hotel (and thus get an infinite tax deduction). He gives his son all the even numbered rooms, physically removing them and putting them on a handy nearby island. Now each person has their own hotel, each one with half the rooms of the original Hilbert Hotel.

After the tax deduction is accepted by the IRS, they each renumber the rooms; the son renumbering each room of his hotel by giving it a new number equal to half the old number, and the father renumbering each of his own rooms by giving it a new number equal to half of (the old number plus one). This of course does not change the size of the hotel, so no new zoning permits are required.

However, each hotel is now a complete and full-fledged Hlibert Hotel. Something from nothing. This works because even the smallest infinity is... very big. This is the essence of the B-T "paradox". B-T does some fancy stuff, but the basics are lodged in the properties of infinity.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

Eebster the Great wrote:The fact that you think the set of points in a circle is finite is already extremely problematic.

A valid criticism. You have told me that I have insufficiently defined finite.

I say that something is finite, if the set(circle) that contains it, is closed as I have defined it. I further state that it is true for all integers, n, equal to or greater than 3. When closed in that fashion, ending upon itself, I bound the circle to a specific value since the end points have no space between them. A condition of the real number line. That may be true for numbers other than integers, but for clarity I omit them.

Does this satisfy your objection?

@ucim I have seen(not watched) a video that puts a tune to it. To paraphrase Orwell.

"My sight is failing," she said finally. "Even when I was young I could not have read what was written there. But it appears to me that that wall looks different. Are the Seven Commandments the same as they used to be, Benjamin?"

For once Benjamin consented to break his rule, and he read out to her what was written on the wall. There was nothing there now except a single Commandment. It ran:

ALL INFINITIES ARE EQUALBUT SOME INFINITIES ARE MORE EQUAL THAN OTHERS

@Sizik I have watched Vsauce's video. You are now watching the Idiot's guide to the Paradox.

Paraphrasing Orwell, morriswalters wrote:ALL INFINITIES ARE EQUALBUT SOME INFINITIES ARE MORE EQUAL THAN OTHERS

:)

Spoiler:

You probably know this, but as it turns out, not all infinities are equal. Some are more than others. Others are equal.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

A circle is bounded and its measure (length) is finite, but the number of points making up the circle is the same as the number of points making up the real number line.

Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.---If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

morriswalters wrote: What I doubt is your skill as a teacher. If you don't want to teach idiots, don't, but please don't tell the idiots to read the book, it isn't very helpful. I'm an idiot, remember?

I doubt you are really that much of an idiot. You're probably even curious about these things, so you should do fine!

An important thing to do when teaching is to determine what to teach. It's an extremely ad hominem process! You are a hominem and I am directing my warm stream of wisdom directly ad you.

My point is, start with definitions. The basics of set theory and point set topology are not very interesting. They are boring and you will not like them. The things you are talking about in this post, the expansion of the universe and banach tarski, these are both rather interesting!

If you want to understand these things though, the definitions really are important. I recommend starting with these (and you don't have to purchase them, they're both made freely available (and not by vexatiously piratical means, but rather by authorial sharing.))

It definitely has a component of "read the book," to be sure, but I am not the only person who has advocated a "flipped" classroom in this model. In fact, it's in quite a bit of vogue right now! I could type out some lectures and then assign homework, but really, it does work so much better if you try a little work on your own first, and come ask when you find something confusing. And not when you find banach tarski confusing, but when you find the formal definitions of open and closed confusing, for example. (And they are confusing! What is the deal with sets that are both closed and open, amirite? Weirdos, for sure.)

And really, ditch the claims to your own idiocy.

As Fr. Sullivan, who taught me the calculus, frequently said: "I could teach AP calculus to a stone, if the stone just did the homework."

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

gmalivuk wrote:A circle is bounded and its measure (length) is finite, but the number of points making up the circle is the same as the number of points making up the real number line.

No. I used that property of the real line as a condition of closure. Each circle has a finite number of points to define each circle. So each circle is finite for any n. For n greater than or equal to three. Is this sufficient?

End replyDiscussion

To be clear let me discuss why n is bounded in this fashion. n=1 isn't defined in this set for a circle, since we defined a circle as two points. n= 2 should work. But it fails the condition of ending where it started. That circle isn't closed. To close it you would have to steal the origin for n=1. That origin is unique. I can't deal with it easily so I left the bounds at n greater than equal to three. You can look at how the point moves along the shape however. It's an ascending spiral with height always=0 for all n, and with all n different than the starting point.

I want the circles to be distinct. They are already identical since they have the same origin. The sets are about book keeping. Since the distinctness is the condition I desire, I defined the circle in a way that made each circle distinct. I believe this shows something of note. But that is an unnecessary and confusing digression. But I'll sketch it without defending it.

I can define the sphere as congruent to the circle by making the definition of the shape, sphere, equal to 1 point and a origin. Moreover when I do that, each circle defines exactly one sphere. More generally this is true for any shape in spherical space. The shape is defined by its surface and not its interior. The origin merely places it at a point in space. No one has yet attacked the geometry. I'm getting feedback on my definitions, but none on my description of movement on the line. Is the model correct in function? Throw it out the window or what?

Morris, you are mistaken, and this is not a debatable point. There are as many points in any (non-degenerate) circle as there are in the entire plane. Give me a circle of any radius and I can give you a one-to-one (and onto) correspondence between points on that circle and points in the plane. There are not only infinitely many points in a circle, there are uncountably many points.

The fact that the circle is a subset of the plane makes this very counterintuitive, but in fact, by definition, every infinite set has a similar property. For example, although the even numbers are a subset of the integers, it is easy to show that for every even number there is exactly one integer. Simply consider the function f(x)=2x.

I think morriswalters is using his own definitions of "finite", "bounded", and "closed", possibly among others, which makes this discussion rather pointless.

Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.---If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

gmalivuk wrote:I think morriswalters is using his own definitions of "finite", "bounded", and "closed", possibly among others, which makes this discussion rather pointless.

Dare I mention Pressures?

morriswalters wrote:...I can define the sphere as congruent to the circle by making the definition of the shape, sphere, equal to 1 point and a origin. Moreover when I do that, each circle defines exactly one sphere.

Well, no. This only works if you've pre-defined a 3-space in which the 2-circle appears. Similarly, if you pre-define a 2-space (plane) in which a 1-circle (two points on a line equidistant from the origin) appears, there is only one circle defined by that same (origin) point and the radius, but if you are thinking in 3-space, then there are infinitely many circles that satisfy this condition; they are embedded in different planes at different angles to each other.

So, to your point, similarly, if you consider 4-space, there are infinitely many 3-spaces embedded in it, each at different angles to each other, which can hold different spheres with the same origin and radius. We can't "imagine" them because we are three-dimensional beings. But it is so nonetheless.

A suggestion - if you want to define words, use new words that don't already have other meanings attached to them. For example, for your definition of "closed", instead use "snaked" (as in "snake eating its tail"). That shields us from the glare of the old definition, and allows us to make use of the new one as you have defined it.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

morriswalters wrote:To be clear let me discuss why n is bounded in this fashion. n=1 isn't defined in this set for a circle, since we defined a circle as two points. n= 2 should work. But it fails the condition of ending where it started. That circle isn't closed. To close it you would have to steal the origin for n=1. That origin is unique. I can't deal with it easily so I left the bounds at n greater than equal to three. You can look at how the point moves along the shape however. It's an ascending spiral with height always=0 for all n, and with all n different than the starting point.

Could you draw some pictures of what you're saying here? I'm not sure how your definition of a circle works.

gmalivuk wrote:

King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.

Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

I have spoilered the image since it is so large. And since I have to at some point, I choose n=post.

Spoiler:

@SisikThis is the Circle of Apollonius. In this image put the A and B on the circle and the P at the origin. The shape can be defined by these distinct points, with one being the origin. My condition of closure, is that at least two point on the circle are are coincident. It follows that this isn't true for n less than three. n=1 defines origin. n=2 defines the shape. n=3 defines the circle. The sum of n=1 and n=2.The problem is occurring since because my circles are piece wise continuous, whereas Ebster is looking at a continuous line. His P is outside the circle, my P is inside. I'll make that explicit in the body. But first a wee joke.

Since everyone but me has a degree, I'm closer to the point.(I think it's funny, anyway) I remember the properties of a circle.

Properties (of a circle)

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.) The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T. All circles are similar. A circle's circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2π and π, respectively.

The circle which is centred at the origin with radius 1 is called the unit circle.

Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points.(that would be that part about n greater or equal to 3, so obviously people aren't reading what I'm writing)

ucim wrote:A suggestion - if you want to define words, use new words that don't already have other meanings attached to them.

I will if you will. You tell mathematicians to do so as well. Until that day, I am doing exactly what they are doing. I'm telling you what exactly I mean when I say finite for instance.

gmalivuk wrote:I think morriswalters is using his own definitions of "finite", "bounded", and "closed", possibly among others, which makes this discussion rather pointless.

You are doing what you always do when I'm involved, making it about me rather than the problem. This is why I laugh myself silly when you use the word ad hominem in a discussion. What ucim and you are saying in effect is, we don't speak Chinese, shut up and speak English. I can't since I don't speak English. It doesn't make what I'm saying wrong, because I say it in Chinese. It merely makes it Chinese. I'm trying to develop a pidgin so we can communicate. Savvy Pidgin? If you don't want to take the time to try and understand, go elsewhere. I'm doing what I can with the tools I own.

Eebster the Great wrote:There are not only infinitely many points in a circle, there are uncountably many points.

I never said anything other than that. I'm not counting points. I'm counting n's.

morriswalters wrote:I'll define a shape and call it a circle.

I will further say, that I can define that shape with 1 point and a origin, if the that shape is closed.

Have I not been clear? Here is an alternative statement of the B-T Paradox, from the Wikipedia entry on paradoxical sets.

The Banach–Tarski paradox is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.

My geometry is a helix. The circle is a degenerate helix. My geometry draws the helix with h=0. If t is the number of turns, then t-1 is the number of possible complete circles you can get out of a slinky of the same size.

It's been apparent to me from day one, that this is about distinctness. The B-T Paradox is a proof in Geometry. The Paradox is, that it is counter intuitive to how you think geometry works. Take the limit of n as n tends to infinity and you end up with a circle of an infinite number of points n. I'm looking at all the sets of points that can make up that circlemake up particular circles, you are looking at all the points that could make up that one circle. The sets are finite if n is finite.

Here what I actually said in my original title text. On any sphere there is a distinct point, n, on that sphere, that can be connected to the origin. And to another sphere with that same point n. The edge of the universe is unseeable if it doesn't share that n. Space outside of the universe is a paradoxical set. It's the last piece of the slinky since it has no n. I had never thought of it in exactly that way until I was introduced to the B-T Paradox.

(Even in the way in which I think you're defining your circle as having a finite number of points to define a circle, BTW, the number of possible points to define a given circle is unlimited, even if only a finite subset of them are needed at any one instance. But I'm not entirely sure which hymn-sheet you're singing from, right now. The words sort of fit to the tune I think you're listening to, but it could also be an entirely different song.)

ucim wrote:A suggestion - if you want to define words, use new words that don't already have other meanings attached to them.

I will if you will. You tell mathematicians to do so as well. Until that day, I am doing exactly what they are doing. I'm telling you what exactly I mean when I say finite for instance.

Mathematicians do not (usually) redefine words that they need the old definitions of, and if they do, it causes problems of understanding. But you are doing that. It is a recipe for confusion. (Not that math can be confusing enough as it is!)

That aside, do you see the relationship between B-T and HH?

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

If you want to experience the model I gave you, buy a Slinky. There are two ways you can look at it. It's either circles or flat wire. Pick one. You can connect the endpoints, or you can cut the helix along a line between the two endpoints without touching either end point.

I make a specific testable statement. Match every circle you cut, with the condition that both ends must have been cut. If you do that you end up with two circles that have two ends that haven't been cut. Rearrange them and you can create one more circle fitting the condition. And are left with one circle with no cut end. A circle with no n. The cut removes n. I'm suggesting this geometric construction models the The B-T Paradox.

Soupspoon wrote:the number of possible points to define a given circle is unlimited, even if only a finite subset of them are needed at any one instance

I subtracted the integers from the real number line. And said that point/piece wise, that line is continuous. So when I say a circle has three points, the circle has exactly 3 points. I effectively, am counting points on the real number line. Are you saying I can't do that? I specifically defined that interval when I just took the integers. Doing it this makes every circle unique for all n. Quit looking for a line and read the proof. His shapes are made of discrete points. Why is this so hard? (actually I understand why it is hard, I'm not fluent in the language Math.)

I'm not going to spend the time to develop the next point. I'm having headaches over this. It is incomplete and may be flawed.

Hilbert's Hotel works because we omit 3 intervals. (0,1),(1,2), and (2,3). When you check in, here is your check in condition. The interval (0,1) exists and must exist. So add 1 carry 1 down the series. It returns the series exactly as it originally was. If you start at three and use the same condition then you end up with 2 different sets. This is the tail of the Slinky.(as an aside, 0 makes the odd an even numbers equal in members doesn't it?) Hilbert's hotel implies more odds than evens.

I can also give you a hint why the smallest number of distinct shapes is 5. Count vertexes. You can only have one vertex per circle/sphere located at the origin. 3,and 4 both have none at the origin.

morriswalters wrote: So when I say a circle has three points, the circle has exactly 3 points. I effectively, am counting points on the real number line. Are you saying I can't do that?

A circle is the locus (set) of points that satisfy a certain criteria (being equidistant from the center). A zero dimensional circle has no meaning because there's no such thing as distance in zero dimensions. A one dimensional circle consists of two points (on a line), each the same distance from a third point (the center) that is in the middle of them. A one dimensional circle is not continuous, and not very interesting.

A two dimensional circle consists of an (uncountable) infinite number of points, each of which satisfies the criterion of being equidistant from the center. It takes only three numbers to identify (define) a circle (the x and y coordinate of the center, and a radius), but those three numbers are not the circle. They are the instructions for creating one. The map is not the territory.

As it turns out, three non-colinear points in a plane can also define a circle, in that there is only one circle (set of infinity points that satisfy the defining criteria) that also contains those points. But those three points are not the circle, any more than two distinct points are the line that they define.

morriswalters wrote:Hilbert's Hotel works because we omit 3 intervals. (0,1),(1,2), and (2,3). When you check in, here is your check in condition. The interval (0,1) exists and must exist. So add 1 carry 1 down the series....

No. The intervals are not of interest - only the (positive) integers are. Those are the room numbers. There is no room "nine and three quarters", whether it leads to Hogwarts or not. All the rooms are labeled as integers.

Spoiler:

You could think of the rooms as "occupying the space" between its label and the next one, but this is not necessary, and adds irrelevant confusion. The rooms could be stacked randomly in a pile; it still works, as long as the doors are properly labeled.

When the hotel is full, all rooms are occupied and there's no unoccupied room, yet it can still accommodate a new guest. You simply put the guest in room 1 into room 2, which bumps the guest in room 2 into room 3, literally ad infinitum. Room 1 is now empty, and available for that new guest.

Spoiler:

Because of this, being an actual guest in such a hotel would be mighty inconvenient! It could also be argued that the practical matter of moving guests leaves at least one guest without a room during the move, which is cheating, and so the move should begin with the highest number room, not the lowest number room, and since there is no highest number room the move can't happen. But that's something for mathematical philosophers (or philosophical mathematicians) to tackle.

Using the same trick, an infinite number of guests can be accommodated by moving each existing guest into the room whose number is double the original room number, leaving all odd numbered rooms empty for that giant tourist bus.

Two infinities are equal if you can establish a one-to-one correspondence between their elements, and n->2n works just fine with an infinite set. In that manner, infinite sets contain themselves (as it were) and can be cloned. The Hilbert Hotel can be cloned by extracting the even numbered rooms and renumbering both resulting "half-hotels". A circle (or ball) can be cloned the same way (except for the problem of the origin, which in the Hilbert Hotel we cleverly avoided by using natural numbers to number the rooms, rather than the heretical "start from zero" that computer programmers have farded us with.)

A circle (or ball) deals with an uncountable infinity, whereas the HH deals with a countable infinity, so a bit more cleverness is required in order to clone the (uncountablely infinite) set of points. That's what the B-T procedure does. (I'll call it a "procedure" because it's not really a paradox; it's merely something that's unintuitive; there's no actual contradiction; and nothing says I can't use more than one semicolon in a sentence.) When you have an infinite number of points (countable or not), you never run out of points, which is what lets you get away with this trickery.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

Morris, you seem to think that you are being totally clear, and people here who don't understand you must not be trying hard enough to understand (or possibly just aren't smart enough). I assure you that your posts are in fact totally opaque and make absolutely no sense to somebody not inside your head. The whole point of using mathematical terminology to communicate mathematical ideas is to be understood.

You say that you are speaking Chinese and we are demanding you speak English. That is fair enough, because if you go to England and start asking strangers questions in Chinese, you can hardly complain if they don't understand you. But it is actually worse than that, because if you spoke a common Chinese language, hundreds of millions of people around the world might understand you, but if you use terminology you just make up yourself without adequately defining it, then no other people anywhere can hope to understand you.

ucim wrote:A circle is the locus (set) of points that satisfy a certain criteria (being equidistant from the center). A zero dimensional circle has no meaning because there's no such thing as distance in zero dimensions.

At this point in time I'm prepared to say that my idea is crazy. I'm having no fun defending it.

Sit at your desk and draw a circle. Get some tracing paper lay it over the top of your circle. On the top layer draw three dots so that one is in the center and the other two fall on the edges of the circle below. That's circle one. Put another blank piece on top a do it again with four dots. And so forth. Those are my circles bound on my circle shape.

In the paradox they explicitly use shapes made of points. I'm trying to tell you why I think that is true. gmalivuk told you that points have extent but no volume. I'm I misstating him? That leads to my circle. It's a zero area circle. And it is exactly like a zero volume sphere. Can you draw any circle that can't contain those points? That is what I mean by zero area.

Look at your definition of a circle.

ucim wrote:A circle is the locus (set) of points that satisfy a certain criteria (being equidistant from the center).

You give the general case. But you've done it by telling me how it's defined, without implicitly choosing a particular radius. You've given the extent. Is this in agreement with the way gmalivuk is using extent?

If we agree on that definition of extent, as I interpret gmalivuk, can I not say that when I am counting the number line, I mean precisely what I say. Counting has extent, but not length. Do you agree?

On HH

The two sets we are talking about aren't congruent. They are both infinite. HH implies this condition. Odds greater than evens. One set is missing 2 evens and one odd. A possible alternative is can be stated as below.

If i=infinity and if HH rooms are full. ThenFor HH's new guest................. 1 +i=i (but we think 1+..........+1=c)HH's rooms have extent and nothing else. c/=i HH's rooms can't be full.(/= is not equal)I state that as the Paradox.Full implies c not i.

Eebster the Great wrote:Morris, you seem to think that you are being totally clear

No, I wish I did think that, then I could be frustrated with you because you can't see it. In the true case, I know that I'm opaque, I'm frustrated with me because I can't make it clear. I'm the Chinese guy who wants to order a latte at Starbucks and can't, because I can't speak English. I know frustration cuts both ways.

It's stings when I make a point and you say I'm missing something, and tell me what it is. But it helps me iterate. My knowledge increases. When I get told my meanings aren't the same ones you hold, I know that already, no one is telling me anything that I don't know. It's only noise.

I am grateful when people respond, and I understand at some point they will walk away, because there isn't any reward in putting up with me. And that is just the way it is.

morriswalters wrote: gmalivuk told you that points have extent but no volume.

I definitely didn't tell anyone that.

If we take "extent" to be something like "measure", which can work in different dimensions, then the following are true:Points have zero extent of any kind.Curves have positive extent in the form of length and zero extent of any other kind. (Lines can contain points with zero extent.)Surfaces have positive extent in the form of area and zero extent in the form of volume. They contain curves with positive length but zero area.Solids have positive extent in the form of volume but zero extent in higher dimensions. They contain surfaces with positive area but zero volume.

Note that I have been careful to say "zero" and not "no". Earlier there was confusion about that distinction. The surface of a sphere is measurable in R3. Its measure is exactly zero. The sets in Banach-Tarski are not measurable. There is no number you can assign to them and say, "this is the volume of this set of points".

Also note that, apart from single zero-dimensional points themselves, the number of points is the same in all these sets I've mentioned, in the sense that there is a one-to-one correspondence between the points of a line and the points on the surface of a sphere, and between the points on the surface of a sphere and the points in the interior, and between the points in the interior and the points in the non-measurable B-T sets.

Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.---If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

morriswalters wrote:At this point in time I'm prepared to say that my idea is crazy. I'm having no fun defending it.

Then let's drop that idea. Meanwhile there's an underlying misunderstanding; do you wish to address this? I'll assume "yes" for now.

morriswalters wrote:Sit at your desk and draw a circle. Get some tracing paper lay it over the top of your circle. On the top layer draw three dots so that one is in the center and the other two fall on the edges of the circle below. That's circle one.

No, those three points are not a circle, at least not as "circle" is understood by mathematicians. Similarly, two points are not a line. Nonetheless, those three points define a circle, just like two distinct points define a line, in that there is only one [circle|line] that passes through those points. If you know those points, you can determine all the other points that make up the shape in question.

But it takes all the other points to actually be the circle.

(Also note that a circle does not include its inside. That would be a disk. Similarly, a sphere does not include its inside; that would be a ball. The word "sphere" is often used to mean "ball"; the formula for volume is often stated as the "volume of a sphere", but it's really the "volume of a ball" or the "volume enclosed by a sphere". Yeah, confusing terminology. Bad mathematicians! Bad! Bad!)

Yes, giving the definition of a circle I've not specified a radius. Upon specifying a radius and a center, I've defined the circle I'm talking about. I can define that circle several ways; "the circle centered at the origin (0,0) and having radius 2" is the same circle as "the circle containing the points (0,2), (0,-2), and (2,0)". This is key to undersanding; the circle defined by three points still contains all the other points satisfying the first definition; it's the same circle.

This post by Eebster the Great is where "extent" was first used. Gmalivuk didn't use it. His er... point... was that shapes are embedded in (n-dimensional) spaces, and lower dimensional shapes have zero thickness in the remaining dimensions. For example, a line has zero thickness; thickness only has meaning in more than one dimensional space. So if you consider a (one dimensional) line embedded in (three dimensional) space, it has zero thickness in two (orthogonal) directions. It has zero extent in those directions.

"Extent" has a more specific mathematical meaning, but this will do for now (especially since I don't happen to know it offhand).

The point being, that by defining a circle (in general) or by defining this circle (giving origin and radius, for example), I have not "given the extent". The extent is implicit in the fact that a circle is a line that curves in upon itself to form an endless loop. Lines have zero extent in all directions except one... that's what makes them one dimensional.

Now, some sets (i.e. cantor dust) have intriguing properties, and the B-T procedure uses these properties, because the decomposition isn't into contiguous pieces but into "interlocking infinite dust", whose properties are non-intuitive. Take a look at the Wiki article on cantor dust for more info on these odd properties. You need to grok this before hitting the B-T procedure.

morriswalters wrote:On HH

The two sets we are talking about aren't congruent.

If by the two sets you mean "the even rooms" and "the odd rooms", then I'm not sure what you mean by "congruent", but they have the same cardinality. That is, they have the same number of elements. There are just as many odd rooms as there are even rooms, and there are just as many even rooms as there are prime-numbered rooms.

"Infinity plus one equals infinity."

morriswalters wrote:(but we think 1+..........+1=c)

If by "c" you mean "the continuum", then no, that is not true. The continuum (the cardinality of the reals) is not the same as (1+...+1), which is ℵ0 (the cardinality of the integers). It is thought that c=ℵ1, but this is not proven.

Whether HH rooms have extent is not relevant; what's important is that they have a room number (which is an integer). (Whether they have extent is only important to the guest, who might like a spacious room or a cozy room, but the desk clerk only cares about which key to give xim).

While I'm here, don't get confused between ℵ, which is a cardinal (indicating quantity), and ω, which is an ordinal (indicating position).

The HH hotel has ℵ0 rooms. They are all labeled by finite integers.

The first "infinite integer" is ω, but the HH has no room labeled ω. No guest can ever go there, because room ω does not exist. There's no room just before ω that has an integer label. You can't get there from here.

The HH can be full, but there's always room for more. That's one of the funny things about infinity. And the B-T procedure uses these funny things about infinity. Best way to get a handle on it is to read up on the funny things infinity does, and get comfortable with them. Then the B-T procedure will make more sense.

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

morriswalters wrote:In the paradox they explicitly use shapes made of points. I'm trying to tell you why I think that is true. gmalivuksomeone said I thought,told you that points have no extent butand no volume. I'm I misstating him it? That leads to my circle. It's a zero no area circle. And it is exactly like a zero no volume sphere. Can you draw any circle that can't contain those points? That is what I mean by zerono area

gmalivuk wrote:Note that I have been careful to say "zero" and not "no". Earlier there was confusion about that distinction. The surface of a sphere is measurable in R3. Its measure is exactly zero. The sets in Banach-Tarski are not measurable. There is no number you can assign to them and say, "this is the volume of this set of points".

I'm not sure if we disagree, since I would say the sphere has no surface. It has only points.

@ucimYeah I should have stated it explicitly. Okay.

Infinity not equal to a constant.

Whatever the various versions of infinity are, or are not, doesn't change basic arithmetic. If I don't know what it is, I know what it ain't.

ucim wrote:There are just as many odd rooms as there are even rooms, and there are just as many even rooms as there are prime-numbered rooms.

There aren't if n is greater than or equal to 3. 1 is odd and 2 and 0 are even. If the internet is right. Which I admit is uncertain. Primes are down two with Even, Odd is a man up on the floor. The Bellman is having kittens.

1 represents the interval between zero and one to respond to something said earlier, and higher dimensions and other spaces are sweet but I'm currently involved in a roof collapse in R3. I'm digging out and ambling down to see the medic. Roofs hurt. I've tempted the RNG too much recently.

♪Welcome to the Hotel AlepHilbert...♪You can check out any time you like,But be prepared for a queue.

Firstly, there isn't (unless there is, but it's not usual) a room zero.Secondly, there are an infinite number of rooms. For half of all 'normal' ns, there are unequal numbers from 1 (or, alternately, 0) that are odd vs even, but when n is infinite there are exactly equal numbers. Like there are equal numbers of prime numbers, a fact that you missed but ought to have noted as even more unintuitive.

morriswalters wrote:Whatever the various versions of infinity are, or are not, doesn't change basic arithmetic.

Ah, but you can't do basic arithmetic with infinity. Basic arithmetic is defined only for numbers. Infinity is not a number. This is where you're tripping.

This is why you can remove elements from an infinite set and end up with the same size set. A set with the same cardinality. A set which, if it were finite, I would say had the same number of elements, but because the set is infinite, I cannot say that, because an infinite set does not have a "number" of elements. We have to count infinitudes differently.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

ucim wrote:Ah, but you can't do basic arithmetic with infinity. Basic arithmetic is defined only for numbers. Infinity is not a number. This is where you're tripping.

No, I was tripping when I got here. I failed arithmetic while I was passing Calculus. It was Network Analysis that drove a dagger into my heart. Curse you FFT's.

@someone somewhereIt's that, you can never leave part, that is scaring me. I'm stuck in sphere space.

I caught the primes but I was on the way out the door. Hilbert Hotels is a nice place, but if they never have zeros they're missing all the fun and if they have infinite rooms it isn't a constant. Having said that, the sets with 0,1,2 are not in Hilbert's Hotel. So when the Bellman checked the rooms against the guest list, he discovered they were missing and had to tell the boss why they couldn't get a room.

But the clever Concierge would rather not have to give the manager bad news, so instead, he used a little quick cleverness. Rooms 0, 1, and 2 unfortunately did not exist, and had been booked, it's true, but it's simple as can be to map the occupants to new rooms - 0 -> 3, 1 ->4, 2 -> 5... and everyone who had a room can still find a place to see, exact amount, even without some unfortunate missing bits!

This was going to be very useful next week when all of the non prime rooms are shut down for painting, but they still have no fewer guests...

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Yes, but then Midnight construction Paradoxically put those room numbers back. You know I shoved a guy into room zero and he didn't hit the floor until room 2001 and Dave the current occupant said he had to hit the Monolith and went into the bathroom. He wouldn't help the guy up and the bathroom had a shortcut to the sequel.

Cardinals actually are often called "numbers," and they are a subset of the surreal numbers. The important thing to realize is just that they are not real numbers. "Number," as a general, unqualified term, is not actually defined in mathematics.

I'm pretty sure a topology was defined on or around the first day of my first topo class.

Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.---If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

Dunno about topology, but tautology was defined on the first day of tautology class. Then we were dismissed and the course was over.

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.

My lessons on Paradoxes never taught anything, that's all I know. And I'm not sure I learnt anything in Quantum Mechanics class, once I found out where it was actually being held. But the introductory course about Rooms was a nightmare...

I blame it on the fact that they all went to Hilbert University and the keep getting lost in a deleted interval somewhere close to to L.

Footnote:

Hotel California will never, ever, sound exactly the same for me again. Some where in the background vocals I hear Hilbert's Hotel now. It's disconcerting.

Footnote 2:

I don't mind looking stupid, I hate feeling stupid. I was so happy about something I thought was new, that I missed the obvious. This set seems novel because of the way it's put, but we see these effects daily, if you played in a bathtub as a child especially, and some ways were more fun than others. /facepalm

I hear ya! But always remember - it's not about being smart, it's about becoming smart. You've started on the road to becoming smart about infinity, and that's no small thing.

(...and we haven't even gotten to the big infinities. )

Jose

Order of the Sillies, Honoris Causam - bestowed by charlie_grumbles on NP 859 * OTTscar winner: Wordsmith - bestowed by yappobiscuts and the OTT on NP 1832 * Ecclesiastical Calendar of the Order of the Holy Contradiction * Please help addams if you can. She needs all of us.